# Open-circuit time constant method

Can anyone explain how Rgd is obtained? I plugged both Ix into Rgd expression but did not get the correct result.

here is my attempt

I cannot proceed. What did I do wrong? How can I get rid of Vx ?

$$R_{gd} = \frac{V_xR'_L}{g_mV_{gs}R'_L-I_xR'+V_x}$$
Use $V_{gs}=-I_xR'$:
$$...=\frac{V_xR'_L}{-g_mR'_LI_xR'-I_xR'+V_x}\cdot\frac{\frac{1}{V_x}}{\frac{1}{V_x}}=...\\$$ note that those pesky $\frac{I_x}{V_x}=\frac{1}{R_{gd}}$. You need to rearrange your equation and get to:
$$R_{gd}\left(1-\frac{R'}{R_{gd}}-\frac{gmR'_LR'}{R_{gd}}\right)=R'_L$$